Discrete State-Space Filter and Method for Processing Asynchronously Sampled Data

ABSTRACT

A discrete state-space filter directly applies a linear transfer function that describes the frequency-domain representation of an IIR filter or control plant to asynchronously sampled data. The discrete state-space technique maps a continuous time transfer function into the discrete state-space filter and stores the states of the filter in a sample-time independent fashion in a discrete state-space vector. The filter states are propagated with the asynchronous time measurements provided with the input data to generate the filtered output.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to processing asynchronously sampled data and,more particularly, to the use of a discrete state-space representationto process asynchronously sampled data.

2. Description of the Related Art

A fundamental assumption in classic digital signal processing algorithmsused for filtering and control compensation is that the digital samplesrepresent a uniform (“synchronous”) sampling of the underlying analogsignal. Ensuring uniform time sampling imposes a burdensome constraintupon the design of the system architecture and processing algorithms.

Asynchronous sampling can be caused at a number of different points in asystem for a variety of reasons. In sensing applications the sensor maylose acquisition and the signal may “drop out” for a period of time.Communication channels likewise may suffer data “dropouts” due totemporary loss of signal. The digital sampling performed by the A/Dconverter can produce an asynchronous sequence for a variety of reasons.First, every A/D that is clocked at a uniform time interval has acertain amount of random phase error, or “jitter”. The amount of jittercan be reduced but at increased cost and power consumption. Second, thesystem controlling the A/D may be asynchronous. For example, a low-costcommercial computer running a non-realtime operating system mayinterfere with the application software by preempting access to thehardware hosting the A/D. This can be overcome with a dedicated systemwith native synchronous capability but at increased cost. Lastly, it maybe desirable to intentionally sample the analog signal asynchronously toadapt the sampling to the properties of the analog signal, e.g. localfrequency content or event based triggering. Compression of the sequencemay also cause samples to “drop out” and alternatively, algorithmscapable of asynchronous signal processing may provide utility foroperating on compressed data without the added steps of expansion andrecompression.

Techniques for handling asynchronous sampling typically fall into one oftwo categories. The first approach is to assume that the samples aresynchronous and spend the resources necessary from signal capture,control to the A/D converters to minimize and error and/or to design theoverall system to tolerate or compensate for any asynchronism. This canbe difficult, expensive and result in lower performance. The secondapproach is to convert the asynchronous data sequence into a synchronousdata sequence. A causal technique extrapolates the amplitude of the nextuniform sample from the existing non-uniform values. A 2-pointextrapolation is computationally very simple but tends to amplify noise.Fixed rate estimation and extrapolation uses a dynamic model such as aKalman Filter to predict the amplitude values. This approach hassomewhat better performance but is more complication. A non-causaltechnique is to require the A/D to oversample the analog signal by atleast 4× and more typically 8× or 16× and than interpolate to a uniformsampling rate. This provides better performance but at a much highercomputational burden due to the oversampling.

Direct processing of asynchronous data could provide benefits of cost,efficiency and performance at the logic circuit level by easing thetolerance on the clock and eliminating the requirement for a globalclock to synchronize all parts of a circuit, at a system level byallowing for distributed asynchronous processing and at an algorithmiclevel by allowing for the sampling to be adapted to the signalproperties.

F. Aeschlimann et al. “Asynchronous FIR Filters: Towards a New DigitalProcessing Chain”, Proc. of the 10^(th) Int. Symp. On AsynchronousCircuits and Systems (ASYNC'04) provides a formulation of theconvolution operator for a Finite-Impulse-Response (FIR) filter forasynchronous sampled data. Aeschlimann provides a hardware-architecturefor the convolution operator and demonstrates that the computationalcomplexity of the asynchronous FIR filter can be far lower than that ofthe synchronous FIR filter provided that the signal statistics are wellexploited.

A typical method to synthesize a synchronous Infinite-Impulse-Response(IIR) filter is to map the filter's continuous-time linear transferfunction in the s-domain into discrete time using the bilineartransformation:

$s = {\frac{2}{\Delta \; t} \cdot \left( \frac{1 - z^{- 1}}{1 + z^{- 1}} \right)}$

Since the filter memory is contained in the delay taps, or z⁻¹ terms,which are dependent on the uniform sampling period Δt, the IIR filter isill-suited for operation with a varying Δt. Other mapping techniquessuch as zero and first order hold suffer likewise. The typical approachis to specify a tolerable “jitter” and design the system to accommodatethe worst case jitter. This can be costly and degrade performance.

In a control theory application, the “plant” of a servo compensator ismodeled by mapping the coefficients of the continuous time lineartransfer function into a continuous state-space representation. In mostpractical systems the plant is actually nonlinear. However, due to thecomplexity of solving nonlinear problems, the plant is typically eitherassumed to be linear or the problem formulation is “linearized” to makeit approximately linearly. The continuous state-space representation ismapped into a discrete state-space representation for the uniformsampling period Δt. These discrete state transition matrix, input andoutput gain matrices and direct gain are computed offline and stored.For each successive input sample, a discrete state-space vector isupdated and the amplitude of the output sample is calculated. Thesecontrol applications typically place very stringent requirements on theuniformity of the sampling period. A certain tolerance may beaccommodated by redesigning the underlying transfer function for theplant. However, ensuring that the performance of the servo compensatoris bounded for some worst case deviation in the sampling period willdegrade the overall performance.

An efficient technique or techniques for performing IIR filtering andcontrol modeling on asynchronously sampled data and, more particularly,for adapting existing linear IIR filter and control plant designs forasynchronously sampled data is needed to reduce cost, improveperformance and increase flexibility of the signal processing systems.

SUMMARY OF THE INVENTION

The present invention provides a method for direct application of alinear transfer function that describes the s-domain representation ofan IIR filter or control plant to asynchronously sampled data. Themethod is particularly useful for adapting existing linear IIR filterand control plan designs for uniformly sampled data to asynchronouslysampled data.

This is accomplished with a discrete state-space representation that isupdated for each sample based on the time measurement for that sample.The described state-space technique maps a continuous time transferfunction into a discrete time filter and stores the states of the filterin a sample-time independent fashion in a discrete state-space vector.The filter states are propagated with the asynchronous time measurementsprovided with the input data to generate the filtered output.

In an embodiment, asynchronously sampled data is processed by mappingthe coefficients of a linear homogeneous frequency-domain transferfunction that represents the IIR filter or control plant into acontinuous state-space representation given by matrices A, B, C and D. Adiscrete state vector X_(k) is defined that stores filter statesindependent of sample-time. For each successive data sample u_(k), adiscrete state transition matrix Φ and a discrete input matrix Γ areupdated from the continuous matrices A and B and a time measurementΔt_(k) of the sample. The discrete state transition matrix Φ defines theextent the previous discrete state vector X_(k-1) will affect thecurrent state vector X_(k) and the discrete input matrix F defines theextent the previous state X_(k-1) is expected to change due to inputdata sample u_(k). The discrete state vector is updated by multiplyingthe previous state vector X_(k-1) by transition matrix Φ and adding theproduct of input matrix Γ multiplied by the amplitude of the sampleu_(k). The state vector X_(k) and the sample amplitude u_(k) aremultiplied by matrices C and D, respectively, and summed to give outputsample y_(k). Matrices Φ and Γ are updated for each sample.

In another embodiment, the discrete time filter includes a coefficientarithmetic unit and a filter core. The coefficient arithmetic unit isconfigured to receive the continuous-time state-space representation ofthe linear frequency-domain transfer function and the time measurementof each successive data sample and to update the discrete statetransition matrix and the discrete input matrix from the continuousstate-space representation and the time measurement of the data sample.The filter core stores filter states sample-time independently in thediscrete state vector. The core is configured to receive the updateddiscrete state transition matrix and the discrete input matrix from theCAU, an output gain matrix and an amplitude of the data sample, toupdate the discrete state vector by propagating the filter states in theprevious state vector with the time measurements within the discretestate transition matrix and summing with the sample amplitude weightedby the discrete input matrix, and to calculate an output data sampleamplitude as a function of the updated discrete state vector weighted bythe output gain matrix and the sample amplitude weighted by theinput-to-output gain matrix.

In a real-time application, the discrete state-space matrices Φ and Γmust be updated quickly. This can be accomplished by simply providingsufficient processing power to compute the matrix exponential, which maybe expensive. Alternately, the computational demands can be reduced byformulating the linear system as a cascade or parallel implementation ofa plurality of s-domain transfer functions that are equivalent to thetransfer function and/or simplifying the matrix exponentiation requiredto directly calculation Φ by using either a 1^(st) or higher order orderTaylor series approximation or a Jordan Canonical Form. Another approachwould be to precalculate and store Φ and Γ for a number of values of Δt.Instead of recalculating the matrices for each sample, the algorithmwould select the closest matrices and use them directly or perform aninterpolation. In yet another approach, calculations of Φ and Γ can beconserved by caching results and monitoring Δt. If the value changes bymore than a specified tolerance, Φ and Γ are recalculated. Otherwise thelast updated matrices or default matrices (e.g. for an expected uniformsampling period) are used to update the state vector.

These and other features and advantages of the invention will beapparent to those skilled in the art from the following detaileddescription of preferred embodiments, taken together with theaccompanying drawings, in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 a and 1 b are diagrams illustrating asynchronously sampled data;

FIG. 2 is a flowchart of a method for processing asynchronously sampleddata using a discrete state-space representation;

FIG. 3 is a block diagram of a hardware implementation of anasynchronous state-space filter;

FIG. 4 is a block diagram of the filter's bus interface;

FIG. 5 is a block diagram of the filter's coefficient arithmetic unitfor a parallel implementation and 1^(st) order approximation;

FIG. 6 is a block diagram of the filter's filter core for the parallelimplementation;

FIGS. 7-10 are plots illustrating the performance of the asynchronousstate-space filter for an IIR filter design; and

FIGS. 11-13 are plots illustrating the performance of the asynchronousstate-space filter for a linear control plant design.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a method for direct application of alinear transfer function that describes the frequency or s-domainrepresentation of an IIR filter or control plant to asynchronouslysampled data. The described discrete state-space technique maps acontinuous time transfer function into a discrete time filter and storesthe states of the filter in a sample-time independent fashion in adiscrete state-space vector. The filter states are propagated with theasynchronous time measurements provided with the input data to generatethe filtered output.

Digital signal processing algorithms used for filtering and controlcompensation include both linear and non-linear problems. Becausenon-linear problems are much more difficult to solve, they areoftentimes either ‘assumed’ to be linear or are ‘linearized’ prior toformulation. Thus, linear systems constitute a substantial majority ofthe practical problems in signal processing applications. Also, thereexists many IIR filter and plant designs for known linear problems. Theability to adapt these for asynchronously sampled data is a greatbenefit.

A linear system is described by a differential equation relating afunction x with its derivatives such that only linear combinations ofderivatives appear in the equation and takes on the general form:

x=A ₁ d(x)+A ₂ d ²(x) . . . A _(n) d ^(n)(x)

where d is the derivative operator and A is a constant. Linearityindicates that no function of x or its derivatives will exceed order onenor include any functions of any other variable. Beyond the mathematicaldefinition, linear systems have the desirable properties of havingunique, existing solutions and of maintaining energy at any discretefrequency at that same frequency. Furthermore a linear system can bedescribed by a cascade of simpler linear systems.

Examples of asynchronously sampled data 10 having amplitude u_(k) attime t_(k) are illustrated in FIGS. 1 a and 1 b. The sampling shown inFIG. 1 a may be representative of a uniform sampling A/D that exhibitssignificant “jitter”, an asynchronous control system such as a low-costcommercial computer running a non-realtime operating system orintentional asynchronous sampling adapted to the analog signalproperties. The sampling shown in FIG. 1 b is representative of auniform sampling A/D that exhibits minimal jitter but is subject to“drop out” from, for example, the acquisition sensor or a communicationchannel. In both cases, the average sampling rate must still satisfy theNyquist criteria. In accordance with the invention, each input samplemust not only include its amplitude u_(k) but also a time measurementt_(k) or Δt_(k).

A method of discrete time filtering asynchronously sampled data todirectly apply a linear transfer function H(s) that describes thes-domain representation of an IIR filter or control plant toasynchronously sampled data u_(k), Δt_(k) is illustrated in FIG. 2. Thefirst step (step 12), which is suitably performed off-line, is to mapthe coefficients of the linear transfer function H(s) given by:

$\begin{matrix}{{H(s)} = \frac{{b_{0}s^{N}\ldots} + {b_{N - 1}s} + b_{N}}{{s^{N}\ldots} + {a_{N - 1}s} + d_{N}}} & (1)\end{matrix}$

into a continuous state-space representation in which,

$\begin{matrix}{{\overset{.}{X} = {{A \cdot X} + {B \cdot u}}}{Y = {{C \cdot X} + {D \cdot u}}}{{where}\text{:}}{A = \begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\0 & 0 & 1 & \cdots & 0 \\\vdots & \vdots & \vdots & ⋰ & \; \\0 & \cdots & 0 & 0 & 1 \\{- a_{N}} & \cdots & {- a_{3}} & {- a_{2}} & {- a_{1}}\end{bmatrix}}{B = \begin{bmatrix}0 \\\vdots \\0 \\1\end{bmatrix}}{{C = \begin{bmatrix}{b_{m} - {b_{0} \cdot a_{m}}} & \cdots & \left( {b_{2} - {b_{0} \cdot a_{2}}} \right) & \left( {b_{1} - {b_{0} \cdot a_{1}}} \right)\end{bmatrix}},{and}}{D = {b_{0}.}}} & (2)\end{matrix}$

and where:

u is the input,

X is the continuous state-space vector,

Y is the output,

A is the state transition matrix,

B is input gain matrix,

C is the output gain matrix, and

D is the input-to-output gain matrix, which is a scalar for a singleinput, a column vector single input/multiple output, a row vectormultiple input/single output, and a matrix for multiple input/multiple.D is often zero.

For the case of a single input channel in which a single input datasample is presented to the discrete state-space filter at a time,matrices B and C default to vectors and vector D defaults to a scalarvalue. If multiple input channels are present and sampled at the sameasynchronous times they may be processed through the same discretestate-space filter.

The discrete state-space filter for asynchronously sampled data isdefined by the following transformation:

$\begin{matrix}{X_{k} = {{\Phi \cdot X_{k - 1}} + {\Gamma \cdot u_{k - 1}}}} & (3) \\{{Y_{k} = {{C \cdot X_{k}} + {D \cdot u_{k - 1}}}}{{where}\text{:}}} & (4) \\{\Phi_{k} = ^{{A \cdot \Delta}\; t_{k}}} & (5) \\{\Gamma_{k} = {\left( ^{{A \cdot \Delta}\; t_{k - 1}} \right) \cdot A^{- 1} \cdot B}} & (6)\end{matrix}$

and where:

X_(k) is the discrete state-space vector that stores the filter statesindependent of sample-time,

Y is the output,

Φ_(k) is the discrete state transition matrix defining the extent theprevious discrete state vector X_(k-1) will affect the current statevector X_(k), and

Γ_(k) is the discrete input matrix defining the extent the previousstate is expected to change due to input data sample u_(k)

The filter memory elements are the states stored in X_(k) and areindependent of Δt. Matrices Φ_(k) and Γ_(k) are updated for eachasynchronous time measurement Δt_(k) to propagate the asynchronousmeasurements in the filter states. The discrete state-space vector isinitialized by, for example, setting all states to zero.

The core steps of the discrete state-space filter will be describedfirst. Thereafter, optional steps and embodiments that improvecomputational efficiency and stability will be revisited.

The sample counter k is set to zero (step 14) and the discretestate-space filter receives an input sample u_(k), Δt_(k) (step 16). Thediscrete state transition matrix Φ_(k) and the discrete input matrixΓ_(k) are updated from the continuous state-space representation(matrices A and B) and the time measurement Δt_(k) of the input datasample (step 18). This “update” can be done in a number of waysincluding a direct calculation of Φ_(k) and Γ_(k) as given in equations5 and 6. The state vector X_(k) is updated according to equation 3 (step20) and the output y_(k) is calculated according to equation 4 (step22). The output amplitude y_(k) occurs at the asynchronous sample timet_(k) or Δt_(k) with respect to the previous sample. The sample counterk is incremented (step 24) and the next sample is processed.

As given in equation 3, the transformation from a continuous state-spacerepresentation to the discrete state-space filter requires a matrixexponentiation to calculate Φ_(k). A direct computation of this matrixexponentiation is computationally burdensome and can cause stabilityproblems, particularly as the filter order “m” gets larger. Round-offerror and finite precision arithmetic causes numerical instability. Thisis particularly problematic for filters that operate with narrowrejection bands. There are a number of techniques for simplifying thecalculation, precalculating and storing the discrete matrices andselectively skipping the update that can be implemented.

The direct calculation can be simplified by formulating the linearsystem H(s) as a cascade or parallel implementation of a plurality of1^(st) and 2^(nd) order s-domain transfer functions H¹(s), H²(s) . . .that are equivalent to the transfer function H(s). There are typicallyN/2 stages if N is rounded up to the nearest even integer. Thisdecomposition is well-known using partial fraction expansion for linearsystems. If each stage is stable than the cascade or parallelimplementation is stable. Each of the transfer functions H¹(s), H²(s) .. . are mapped into a continuous state-space representation andtransformed into the discrete state-space filter as shown in FIG. 2. Theparallel implementation is faster but requires more parallel processingcapability. Another simplification, which can be done for the entirelinear system H(s) or each of the stages H¹(s), H²(s) . . . of acascade/parallel implementation, is to compute Φ_(k) using either a1^(st) order approximation or a Jordan Canonical Form.

The 1^(st) order approximation is given by:

$\begin{matrix}{{\Phi_{k} = {I + {{A \cdot \Delta}\; t_{k}}}}{\Gamma_{k} = \begin{bmatrix}0 \\\vdots \\{\Delta \; t_{k}}\end{bmatrix}}} & (7)\end{matrix}$

where I is the identity matrix. This 1^(st) order approximation workswell when the linear system is decomposed into a cascade/parallelimplementation of 1^(st) and 2^(nd) order functions.

The Jordan Canonical Form simplifies the matrix exponentiation to ascalar exponentiation and is given by:

$\begin{matrix}{{J = {V^{- 1} \cdot A \cdot V}}{Where}} & (8) \\{J = \begin{bmatrix}\lambda_{1} & 0 & 0 \\0 & ⋰ & 0 \\0 & 0 & \lambda_{m}\end{bmatrix}} & (9)\end{matrix}$

And λ=eigenvalue of A., and

$\begin{matrix}{{\Phi^{\prime} = \begin{bmatrix}^{\lambda_{1}\Delta \; t} & 0 & 0 \\0 & ⋰ & 0 \\0 & 0 & ^{{\lambda_{m} \cdot \Delta}\; t}\end{bmatrix}}{\Gamma^{\prime} = {\left( {\Phi^{\prime} - I} \right) \cdot V \cdot B}}{C^{\prime} = {C \cdot V^{- 1}}}{D^{\prime} = D}} & (10)\end{matrix}$

The Jordan Canonical Form is a simplification of the matrixexponentiation but the output samples y_(k) will have the same value.This approach would be preferred if a parallel/case implementation wasnot used. The Jordan Canonical Form is more efficient for thecomputation of higher order systems.

Another approach would be to precalculate and store Φ and Γ for a numberof values of Δt (step 26). Instead of recalculating the matrices foreach sample, the filter would select the closest matrices and use themdirectly or perform an interpolation. This would sacrifice some amountof accuracy in exchange for speed. In yet another approach, calculationsof Φ and Γ can be conserved by caching results (step 28) and monitoringΔt (step 30). If the value changes by more than a specified tolerance, Φand Γ are recalculated (step 18). Otherwise the last updated matrices ordefault matrices (e.g. for an expected uniform sampling period) are readout of the cache (step 28) and used to update the state vector. Thelatter approach is particularly efficient for systems that are designedto be sampled at uniform intervals but for some reason experience “dropout”. A large percentage of the samples can be processed with the cachedor “default” matrices. When a drop out is detected resulting in a Δtdifferent from that in cache, the matrices are updated.

A hardware implementation of a single input, single output discretestate-space filter 50 that uses a 1st order approximation of the matrixexponential for a parallel form of 2nd order subsystems is illustratedin FIGS. 3-6. Filter 50 includes two primary components; a coefficientarithmetic unit (CAU) 52 that updates Φ_(k) and Γ_(k) and a filter core54 that updates the state vector X_(k) and calculates the output y_(k).

The continuous state-space representation matrices A, C and D areprovided from a coefficient bus 56 to a bus decoder 58. In thisimplementation, matrix B is [0, 0, . . . 1] and is hardcoded into theCAU. Matrices A, C and D are expressed in the appropriate parallel formin which sub-matrix A is 2×2, C is 2×1 and D is a scalar. There are N/2sets of sub-matrices where the index M=N−1. Bus decoder 58 decodes theincoming data and writes the sub-matrices into the appropriate locationsin a static coefficient memory 60. The continuous state-transitionsub-matrix A is directed to the CAU 52 and the output gain andinput-to-output sub-matrices C and D are directed to the fitter core 54.

For each input data sample, CAU receives time measure Δt_(k) and updatesΦ_(k) and Γ_(k). As shown in FIG. 5, each of the N/2 stages 61calculates a 1^(st) order approximation of the matrix exponential asgiven by equation 7 above. Higher order approximations can beimplemented at the cost of increased processing capability. Morespecifically, Δt_(k) is muliplied by each of the four static Asub-matrix coefficients 62 (e.g. A₁₁, A₁₂, A₂₁ and A₂₂) and a scalarvalue 1 is added to the diagonal terms A₁₁ and A₂₂ to give the fourdynamic Φ_(k) sub-matrix coefficients 64. Γ₁=0 and Γ₂=Δt_(k). Theupdates Φ_(k) and Γ_(k) sub-matrices are directed to the filter core 54.

As described above, a cache controller 66 may be included to monitorΔt_(k) and decide whether a recalculation of Φ_(k) and Γ_(k) iswarranted or whether the previously updated sub-matrices Φ_(k-1) andΓ_(k-1) or default sub-matrices, which were stored in a coefficientcache 68, are acceptable. In a system that is ordinarily synchronous butsubject to drop-out this approach can conserve considerable processingresources. Although not shown in this embodiment, Φ and Γ sub-matricesfor Δt could be read in from a bus and stored in memory. The controlleror CAU (if reconfigured) could access the memory to select theappropriate sub-matrices for each Δt_(k).

Filter core 54 receives the continuous sub-matrices C and D, the updatedΦ_(k) and Γ_(k) submatrices and the amplitude u_(k) of the currentsample, updates the state vector X_(k) as given by equation 3 andcalculates the output y_(k) as given by equation 4. More specifically asshown in FIG. 6, to update the state vector X_(k)=[X₁,X₂] for the firststage 61 the sample amplitude u_(k) is multiplied by Γ_(k)=[ΓΓ₁,Γ₂] andsummed with the product of Φ_(k)=[Φ₁₁, Φ₁₂, Φ₂₁, Φ₂₂] and the previousstate-vector X_(k-1)=[X₁,X₂]. The updated state vector X_(k)=[X₁,X₂] ismultiplied by output gain sub-matrix C=[C₁,C₂] and summed to provide ascalar value for C*X_(k). In this implementation the scalar values foreach stage are summed together and than added to the produce of theinput-to-output gain D multiplied by sample amplitude u_(k) to providethe filtered output sample y_(k). In many instances D is zero.

The performance of the discrete state-space filter has been simulatedfor both IIR notch filters and a linear control plant as part of a servocompensator and the results compared to classic IIR filters and controlplants for both uniformly and asynchronously sampled data. As will beshown, the discrete filter's performance is the same or even better thanthe classic techniques for uniformly sampled data and far superior forasynchronously sampled data. The discrete filter does not suffer fromthe frequency domain warping inherent in classic mapping techniques suchas the zero-order hold or bilinear transform. Furthermore, the discretefilter is superior to over-sampling because it does not injectinterpolation or extrapolation error. The discrete filter inherently andautomatically adapts to the type and extent of asynchronism in thesampled data and thus enables the use of less expensive, lower power A/Dconverters and the use of low-cost computers and operating systems forreal-time control computing problems. The cost of this improvedperformance is approximately 6× computational complexity of anequivalent order IIR filter

The performance of the discrete state-space filter and a classic IIRfilter for a variety of test conditions are illustrated in FIGS. 7-10.The modeled filter is a 2-pole notch with 30 dB of rejection at 60 hz.The input signal 100 includes a 0 dB signal 102 at 90 hz withinterference of 0 dB at 60 hz 104 +/−40 dB Gaussian noise 106 as shownin FIG. 7 a. In the first test, the performance of the classic IIR notchfilter and discrete filter were evaluated assuming ideal uniformsampling of Δt=1 ms. As shown in FIGS. 7 b and 7 c, the filtered spectra108 and 110 for the classic IIR notch filter and the discrete filter,respectively, and virtually identical. Both filters pass the desiredsignal 102 and provide about −30 dB rejection of the 60 hz interference104. Note however that this test allowed the classic IIR filtered to bedesigned for ideal uniformly sampled data. If the IIR filter wasdesigned to tolerate some amount of jitter there would be noticeableperformance degradation even if the actual sampled data were perfectlyuniform. The designed in tolerance creates some performance degradationof the filter and cannot adapt to changing sampling conditions. Likewisethe discrete filter is based on the existing transfer function for theIIR filter assuming ideal uniformly sampled data but adapts to thechanging sampling conditions.

In a second test, the input signal 100 was asynchronously sampled atΔt=1 ms+/−0.5 ms of jitter with a uniform distribution. The increase inthe noise floor 112 of the input signal shown in FIG. 8 a is a result ofan artifact of non-uniform sampling and the FFT used for analysis. Asshown in FIG. 8 b, the filtered spectra 114 of the classic IIR rejectsonly 12 dB of the interference 104. By comparison, the filtered spectra116 of the discrete state-space filter rejects 26 dB of the interference104. The discrete filter adapts to the changing sampling conditions ateach sample, hence is able to approximately maintain the designedperformance levels.

In a third test, the input signal 100 shown in FIG. 9 a was uniformlysampled at Δt=1 ms but with a 25% drop-out rate creating asynchronousdata. As shown in FIG. 8 b, the filtered spectra 118 of the classic IIRrejects only 3 dB of the interference 104. By comparison, the filteredspectra 120 of the discrete state-space filter rejects 23 dB of theinterference 104. The discrete filter adapts to the changing samplingconditions caused by the drop-out, hence is able to largely maintain thedesigned performance levels.

In a fourth test, the modeled filter is the same 2-pole notch with 30 dBof rejection at 60 hz. The input signal 130 includes a 0 dB signal 132at 90 hz and a 0 dB signal 134 at 490 hz with interference of 0 dB at 60hz 136 +/−40 dB Gaussian noise 138 as shown in FIG. 10 a. For thediscrete filter, the signal is asynchronously sampled at Δt=1 ms+/−0.5ms of jitter with a Gaussian distribution. As shown in FIG. 10 c, thefiltered spectra 140 provides approximately 20 dB of rejection at 60 hzand passes the signal at 90 hz and 490 largely unaffected. For theclassic IIR filter, the signal is resampled by 4× at Δt=0.25 ms and thaninterpolated to create uniformly sampled data. As shown in FIG. 10 b,the filtered spectra 142 provides about 15 dB of rejection at 60 hz,passes the signal at 90 hz but attenuates the higher frequenciesincluding the signal 134 at 490 hz.

The performance of the discrete state-space filter and a classic IIRcompensator for a linear control plant under a variety of testconditions are illustrated in FIGS. 11-13. The modeled linear controlplant is a double integrator servo.

In the first test, the performance of the classic IIR compensator andthe discrete state-space compensator were evaluated assuming idealuniform sampling of Δt=1 ms. As shown in FIG. 11, the impulse responses150 and 152 of the IIR compensator and state-space compensator,respectively, are virtually identical. However, as in the notch filterapplication, if the classic IIR compensator were designed to toleratesome specified amount of jitter by modifying its transfer function theperformance would be degraded even if the actual sampling were ideallyuniform. Conversely, the state-space compensator can be designed fromthe ideal transfer function, no built in tolerance is required.

In a second test, the input signal was asynchronously sampled at Δt=1ms+/−0.5 ms of jitter with a uniform distribution. As shown in FIG. 11,the step responses 160 and 162 of the IIR compensator and state-spacecompensator, respectively, were plotted for one-hundred runs of a MonteCarlo simulation. The simulation revealed a wide variation in thetransient response of the classic IIR compensator and minimal variationin the state-space compensator.

In a third test, the input signal was uniformly sampled at Δt=1 ms butwith a 50% drop-out rate creating asynchronous data. As shown in FIG.12, the impulse responses 170 and 172 of the IIR compensator andstate-space compensator, respectively, were plotted for one-hundred runsof a Monte Carlo simulation. The simulation again revealed a widevariation in the transient response of the classic IIR compensator andminimal variation in the state-space compensator.

While several illustrative embodiments of the invention have been shownand described, numerous variations and alternate embodiments will occurto those skilled in the art. Such variations and alternate embodimentsare contemplated, and can be made without departing from the spirit andscope of the invention as defined in the appended claims.

1. A method of discrete time filtering asynchronously sampled data,comprising: a) Mapping the coefficients of a linear frequency-domaintransfer function H(s) into a continuous-time state-spacerepresentation; b) Initializing a discrete state vector X_(k) thatstores filter states independent of sample-time; c) Receiving anamplitude u_(k) and time measurement Δt_(k) of at least one input datasample; d) Updating a discrete state transition matrix Φ_(k) and adiscrete input matrix Γ_(k) from the continuous state-spacerepresentation and the time measurement Δt_(k) of the input data sample,said discrete state transition matrix Φ_(k) defining the extent theprevious discrete state vector will affect the current state vector andsaid discrete input matrix Δ_(k) defining the extent the previous filterstate is expected to change due to input data sample; e) Updating thediscrete state vector X_(k) by propagating the filter states in theprevious state vector X_(k-1) with the time measurements within thediscrete state transition matrix Φ_(k) and summing with the sampleamplitude u_(k) weighted by the discrete input matrix Γ_(k); f)Calculating at least one output data sample amplitude y_(k) from theupdated discrete state vector X_(k), the sample amplitude u_(k) andcontinuous-time state-space representation; and g) Repeating steps cthrough f for the next asynchronous data sample.
 2. The method of claim1, wherein the linear frequency-domain transfer function is an infiniteimpulse response (IIR) filter.
 3. The method of claim 2, wherein the IIRfilter transfer function is the transfer function for an existing designassuming ideal uniform sampling.
 4. The method of claim 1, wherein thelinear frequency-domain transfer function is a linear control plant. 5.The method of claim 4, wherein the control plant transfer function isthe transfer function for an existing design assuming ideal uniformsampling.
 6. The method of claim 1, wherein Φ_(k) and Γ_(k) are updatedaccording to calculations Φ_(k)=e^(A)*^(Δtk) and Γ_(k)=(Φ_(k)−1)*A⁻¹*Bwhere I is the identity matrix, A is the continuous state transitionmatrix and B is the input gain matrix from the continuous-timestate-space representation.
 7. The method of claim 6, wherein Φ_(k) andΓ_(k) are updated using a first order approximation of the matrixexponential e^(A)*^(Δtk).
 8. The method of claim 6, wherein Φ_(k) andΓ_(k) are updated using a Jordan Canonical Form to simplify the matrixexponential e^(A)*^(Δtk).
 9. The method of claim 1, whereinX_(k)=Φ_(k)*X_(k-1)+Γ_(k)*u_(k).
 10. The method of claim 1, whereiny_(k)=C*X_(k)+D*u_(k) where C is the output gain matrix and D is theinput-to-output gain matrix from the continuous-time state-spacerepresentation.
 11. The method of claim 1, further comprising: Usingpartial fraction expansion to decompose the linear frequency-domaintransfer function H(s) into a plurality of transfer functions; andPerforming steps (a)-(g) for each said transfer function in parallel orin cascade.
 12. The method of claim 1, further comprisingpre-calculating and storing Φ and Γ for a plurality of Δt values,wherein Φ_(k) and Γ_(k) are updated by reading out Φ and Γ in accordancewith the value of Δt_(k).
 13. The method of claim 12, further comprisinginterpolating Φ_(m) and Γ_(m) at Δt_(m) and Φ_(n) and Γ_(n) at Δt_(n)where Δt_(m)<Δt_(k)<Δt_(n) to update Φ_(k) and Γ_(k).
 14. The method ofclaim 1, further comprising: Monitoring Δt_(k) received in step (c); IfΔt changes by more than a specified tolerance, updating Φ_(k) and Γ_(k)in step (d); and Otherwise skipping step (d) and using the last updatedmatrices Φ_(k-1) and Γ_(k-1) or default matrices Φ_(U) and Γ_(U) for anexpected uniform sampling period to update the state vector X_(k) instep (e).
 15. The method of claim 14, wherein the data samples areordinarily uniformly sampled but subject to drop out that causes Δt_(k)to be asynchronous, said default matrices Φ_(U) and Γ_(U) being used toupdate the state vector for the uniformly sampled data samples.
 16. Amethod of processing asynchronously sampled data, comprising: a) Mappingthe coefficients of a linear frequency-domain transfer function H(s)into a continuous-time state-space representation including a statetransition matrix A, an input gain matrix B, an output gain matrix C anda input-to-output gain matrix D; b) Receiving an amplitude u_(k) andtime measurement Δt_(k) of a k^(th) data sample; c) Updating a discretestate transition matrix Φ_(k)=e^(A)*^(Δtk) and a discrete input matrixΓ_(k)=(Φ_(k)−I)*A⁻¹*B where I is the identity matrix; d) Updating adiscrete state vector X_(k)=Φ_(k)*X_(k-1)+Γ_(k)*u_(k); e) Calculating anoutput amplitude y_(k)=C*X_(k)+D*u_(k); and f) Repeating steps b throughe for the next k+1 asynchronous data sample.
 17. The method of claim 16,wherein the linear frequency-domain transfer function is an infiniteimpulse response (IIR) filter for an existing design assuming idealuniform sampling.
 18. The method of claim 16, wherein the linearfrequency-domain transfer function is a linear control plant for anexisting design assuming ideal uniform sampling.
 19. The method of claim16, wherein Φ_(k) is updated using a first order approximation of thematrix exponential e^(A)*^(Δtk).
 20. A method of using a discretestate-space representation of a linear transfer function including adiscrete state transition matrix Φ and a discrete input matrix Γ toupdate a discrete state vector X_(k) and calculate a output sample y_(k)where Φ_(k) defines the extent the previous discrete state vector willaffect the current state vector and Γ_(k) defines the extent theprevious filter state is expected to change due to input data sample,comprising for each k^(th) input data sample u_(k) having a timemeasurement Δt_(k) Updating the discrete state transition matrixΦ_(k)=e^(A)*^(Δtk) where A is a state transition matrix of thecontinuous-time state-space representation of the linear transferfunction; Updating the discrete input matrix Γ_(k)=(Φ_(k)−I)*A⁻¹*B whereI is the identity matrix and B is the input gain matrix of thecontinuous-time state-space representation of the linear transferfunction; and Updating the discrete state vectorX_(k)=Φ_(k)*X_(k-1)+Γ_(k)*u_(k).
 21. The method of claim 20, whereinΦ_(k) is updated using a first order approximation of the matrixexponential e^(A)*^(Δtk).
 22. A discrete time filter for processingasynchronously sampled data, comprising: A coefficient arithmetic unit(CAU) configured to receive a portion of a continuous-time state-spacerepresentation of a linear frequency-domain transfer function H(s) and atime measurement Δt_(k) of each successive data sample and to update adiscrete state transition matrix Φ_(k) and a discrete input matrix Γ_(k)from the continuous state-space representation and the time measurementof the data sample, said discrete state transition matrix defining theextent the previous discrete state vector X_(k-1) will affect thecurrent state vector X_(k) and said discrete input matrix defining theextent the previous state is expected to change due to input datasample; and A filter core that stores filter states sample-timeindependently in a discrete state vector X_(k), said core configured toreceive the updated discrete state transition matrix Φ_(k) and thediscrete input matrix Γ_(k) from the CAU, an amplitude u_(k) of the datasample and another portion of the continuous-time state-spacerepresentation, to update the discrete state vector X_(k) by propagatingthe filter states in the previous state vector X_(k-1) with the timemeasurements within the discrete state transition matrix Φ_(k) andsumming with the sample amplitude u_(k) weighted by the discrete inputmatrix Γ_(k), and to calculate an output data sample amplitude y_(k) asa function of the updated discrete state vector X_(k), the sampleamplitude u_(k) and the another portion of the continuous-timestate-space representation.
 23. The discrete time filter of claim 22,further comprising: A coefficient memory configured to receive and storethe continuous-time state-space representation including a statetransition matrix A and an input gain matrix B that are directed to theCAU and an output gain matrix C and a input-to-output gain matrix D thatare directed to the filter core.
 23. The discrete time filter of claim23, wherein the CAU updates Φ_(k)=e^(A)*^(Δtk) and Γ_(k)=(Φ_(k)−I)*A⁻¹*Bwhere I is the identity matrix and the filter core updatesX_(k)=Φ_(k)*X_(k-1)+Γ_(k)*u_(k) and calculates y_(k)=C*X_(k)+D*u_(k).24. The discrete time filter of claim 24, wherein the CAU updates Φ_(k)using a first order approximation of the matrix exponentiale_(A)*^(Δtk).
 25. The discrete time filter of claim 22, wherein Φ and Γfor a plurality of Δt values are pre-calculated and stored in acoefficient cache, said CAU updating Φ_(k) and u_(k) by reading out Φand Γ in accordance with the value of Δt_(k).
 26. The discrete timefilter of claim 22, wherein last updated matrices Φ_(k-1) and Γ_(k-1) ordefault matrices Φ_(U) and Γ_(U) for an expected uniform sampling periodare stored in a coefficient cache the further comprising a cachecontroller that monitors Δt_(k). if Δt_(k) changes by more than aspecified tolerance, the cache controller directs the CAU to updateΦ_(k) and Γ_(k) and otherwise directs the CAU to forward the lastupdated matrices Φ_(k-1) and Γ_(k-1) or default matrices Φ_(U) and Γ_(U)from the coefficient cache to the filter core to update the state vectorX_(k).